Free Vibration Analysis of Shafts on Resilient Bearings Using Timoshenko Beam Theory

1999 ◽  
Vol 121 (2) ◽  
pp. 256-258 ◽  
Author(s):  
S. Karunendiran ◽  
J. W. Zu
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Frequency Equation ◽  
Timoshenko Beam Theory ◽  
Mode Shapes ◽  
Complex Eigenvalues ◽  
First Time

This paper presents an analytical method adopted for the free vibration analysis of a shaft, both ends of which are supported by resilient bearings. The shaft is modeled by Timoshenko beam theory. Based on this model exact frequency equation to calculate complex eigenvalues is derived and presented in complex compact form for the first time. Explicit expressions to compute the corresponding mode shapes are also presented.

scholarly journals Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

2011 ◽  
Vol 18 (5) ◽  
pp. 709-726 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Structural Elements ◽  
Mass System ◽  
Assembly Technique

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

Free Vibration Analysis of a Beam Under Axial Load Carrying a Mass-Spring-Mass

2012 ◽  
Author(s):  
O. R. Barry ◽  
Y. Zhu ◽  
J. W. Zu ◽  
D. C. D. Oguamanam
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Axial Load ◽  
Equations Of Motion ◽  
Tensile Load ◽  
Free Vibration Analysis ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Mass System ◽  
Mass Spring

This paper deals with the free vibration analysis of a beam subjected to an axial tensile load with an attached in-span mass-spring-mass system. The equations of motion are derived by means of the Hamilton principle and an explicit expression of the frequency equation is presented. The formulation is validated with results in the literature and the finite element method. Parametric studies are done to investigate the effect of the axial load, the magnitude and location of the mass-spring-mass system on the lowest five natural frequencies and mode shapes. The results indicate that the fundamental mode is independent of the tension and the in-span mass. However, a significant change in all modes is observed when the position of the mass-spring-mass is varied.

Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

2021 ◽  
pp. 108128652110312
Author(s):  
Hooman Danesh ◽  
Mahdi Javanbakht
Keyword(s):  
Free Vibration ◽  
Elasticity Theory ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Two Phase ◽  
Softening Effect

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

scholarly journals An Exact Solution to the Free Vibration Analysis of a Uniform Timoshenko Beam Using an Analytical Approach

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Winkler Foundation ◽  
Closed Form Solutions ◽  
Stiffness Matrices

This study presents an analytical solution to the free vibration analysis of a uniform Timoshenko beam. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. A material law combining bending, shear, curvature, and natural frequency is presented. This complete study is based on this material law and closed-form solutions are found. The free vibration response of single-span systems, as well as that of spring-mass systems, is analyzed. Closed-form formulations of matrices expressing the boundary conditions are presented; the natural frequencies are determined by solving the eigenvalue problem. First-order dynamic stiffness matrices in local coordinates are determined. Finally, second-order analysis of beams resting on an elastic Winkler foundation is conducted.

scholarly journals A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

2017 ◽  
Vol 39 (4) ◽  
pp. 315-328
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Crack Depth ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Frequency Equation ◽  
Form Solution ◽  
Mode Shapes ◽  
Timoshenko Beams

A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

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